Irrationality Sequence

A sequence of Positive Integers $\{a_n\}$ such that $\sum 1/(a_nb_n)$ is Irrational for all integer sequences $\{b_n\}$. Erdös showed that $\{2^{2^n}\}$ is an irrationality sequence.

References

Guy, R. K. ``Irrationality Sequence.'' §E24 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 225, 1994.